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Keywords:
fixed point method; matrix polynomial; matrix differential equation
Summary:
Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.
References:
[1] Davis, G. J.: Numerical solution of a quadratic matrix equation. SIAM J. Scient. Computing 2 (1981), 164-175. DOI 10.1137/0902014 | MR 0622713 | Zbl 0467.65021
[2] Dennis, E., Traub, J. F., Weber, R. P.: On the Matrix Polynomial, Lambda-Matrix and Block Eigenvalue Problems. Computer Science Department, Technical Report, Cornell University, Ithaca, New York and Carnegie-Mellon University, Pittsburgh, Pennsylvania (1971).
[3] Dennis, J. E., Traub, J. F., Weber, R. P.: The algebraic theory of matrix polynomials. SIAM J. Numer. Anal. 13 (1976), 831-845. DOI 10.1137/0713065 | MR 0432675 | Zbl 0361.15013
[4] Dennis, J. E., Traub, J. F., Weber, R. P.: Algorithms for solvents of matrix polynomials. SIAM J. Numer. Anal. 15 (1978), 523-533. DOI 10.1137/0715034 | MR 0471278 | Zbl 0386.65012
[5] Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York (1982). MR 0662418 | Zbl 0486.15008
[6] Higham, N. J., Kim, H. M.: Solving a quadratic matrix equation by Newton's method with exact line searchers. SIAM J. Matrix Anal. Appl. 23 (2001), 303-316. DOI 10.1137/S0895479899350976 | MR 1871314
[7] Higham, N. J., Kim, H. M.: Numerical analysis of a quadratic matrix equation. IMA J. Numer. Anal. 20 (2000), 499-519. DOI 10.1093/imanum/20.4.499 | MR 1795295 | Zbl 0966.65040
[8] Holmes, R. B.: A formula for the spectral radius of an operator. Am. Math. Mon. 75 (1968), 163-166. DOI 10.2307/2315890 | MR 0227783 | Zbl 0156.38202
[9] Kratz, W., Stickel, E.: Numerical solution of matrix polynomial equations by Newton's method. IMA J. Numer. Anal. 7 (1987), 355-369. DOI 10.1093/imanum/7.3.355 | MR 0968530 | Zbl 0631.65040
[10] Lancaster, P.: Lambda-Matrices and Vibrating Systems. Pergamon Press, New York (1966). MR 0210345 | Zbl 0146.32003
[11] Lancaster, P.: A fundamental theorem on lambda matrices with applications II. Difference equations with constant coefficients. Linear Algebra Appl. 18 (1977), 213-222. DOI 10.1016/0024-3795(77)90052-0 | MR 0485917 | Zbl 0388.15004
[12] Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edition. Academic Press, New York (1985). MR 0792300
[13] Pereira, E., Vitória, J.: Deflation of block eigenvalues of block partitioned matrices with an application to matrix polynomials of commuting matrices. Comput. Math. Appl. 42 (2001), 1177-1188. DOI 10.1016/S0898-1221(01)00231-0 | MR 1851235
[14] Pereira, E., Serodio, R., Vitória, J.: Newton's method for matrix polynomials. Int. J. Math. Game Theory Algebra 17 (2008), 183-188. MR 2353584 | Zbl 1177.65065
[15] Shih, M., Wu, J.: Asymptotic stability in the Schauder fixed point theorem. Stud. Math. 2 (1998), 143-148. DOI 10.4064/sm-131-2-143-148 | MR 1636415 | Zbl 0924.47044
[16] Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43 (2001), 235-286. DOI 10.1137/S0036144500381988 | MR 1861082 | Zbl 0985.65028
[17] Tsai, J. S. H., Shieh, L. S., Shen, T. T. C.: Block power method for computing solvents and spectral factors of matrix polynomials. Comput. Math. Appl. 16 (1988), 683-699. DOI 10.1016/0898-1221(88)90004-1 | MR 0973957
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